饱和非线性薛定谔方程多辛Euler-Box方法

对饱和非线性薛定谔方程构造了两个Euler—box格式并将它们组合成了一个新的多辛离散格式．利用新的多辛离散格式模拟饱和非线性薛定谔方程．数值结果表明新的多辛离散格式能够很好地模拟饱和非线性薛定谔方程中孤子波的演化行为,并能近似地保持系统的模平方守恒特性．     

基于非线性薛定谔方程的时间序列去噪

针对噪声污染严重的时间序列,提出一种基于动态随机共振效应的去噪方法.通过具体分析非线性光学系统中的动态随机共振效应,推导出非线性薛定谔方程中的非线性扰动项,得到用于描述非线性系统中的信号传播模型,并应用于时间序列去噪.首先将归一化的时间序列信号输入模型系统,然后通过自适应粒子群优化算法确定系统传播方程中的各项参数,最后...     

求解非线性Schroedinger方程的一个线性化紧致差分格式

通过对非线性项的局部外推,对非线性Schroedinger方程给出了一个线性化紧致差分格式,运用不动点定理和能量方法证明了格式的唯一可解性,文章还运用能量方法和数学归纳法,避开困难的先验估计,证明格式在空间方向和时间方向分别具有四阶和二阶精度,数值算例验证了格式的精度和数值稳定性．     

干扰与控制非同位一维薛定谔方程的输出跟踪

本文研究了干扰与控制非同位一维薛定谔方程的输出跟踪.首先,利用系统的无穷维结构与输出设计了用于估计干扰的无穷维干扰估计器.其次,建立自适应伺服机制使得跟踪误差(u)(1,t)∈L2(0,∞),且闭环系统的所有子系统有界.最后,对闭环系统进行数值模拟,模拟结果表明控制方案的有效性.     

数值连续法解非线性差分方程的可行性

解非线性偏微分方程数值解问题通常可归结为解非线性差分方程组，解非线性方程组的数值连续法是扩大给定方法收敛域的一种尝试。本文正是利用这种方法研究了非线性二阶偏微分方程第一类边值问题数值解的计算问题，并给出检验其算法为可行的充分条件。     

同伦扰动法求解分数阶非线性扩散方程近似解

将Caputo分数阶微分算子引入到带有初值条件的扩散方程中,建立了时空分数阶方程。利用同伦扰动法并借助于Mathematica软件的符号计算功能,求解了分数阶非线性扩散方程的近似解,整数阶方程的结果作为特例被包含。     

求一类非线性偏微分方程解析解的简洁构造算法

通过引入一个变换,利用齐次平衡原理和选准一个待定函数来构造求解一类非线性偏微分方程解析解的算法.作为实例,我们将该算法应用到了mKdV方程,KdV-Burgers方程和KdV-Burgers-Kuramoto方程.借助符号计算软件Mathematica获得了这些方程的解析解.不难看出,该方法不仅简洁,而且有望进一步扩展.     

求解相容方程的Hopfield神经网络

文中提出的改进Ｈｏｐｆｉｅｌｄ神经网络（ＭＨＮＮ），进一步简化了［１］中的结果，除了能求解线性与非线性方程之外，还能求解线性、非线性相容方程，并得到相容方程的最小范数解。     

一类高度非线性网格生成方程的迭代求解方法

在[1]中第170页指出,对于变分方法导出的一类高度非线性的守恒型网格生成方程,采用通常的Picard迭代方法无法正确求解.本文构造了一种新的Picard迭代求解方法,数值结果表明这一方法较好地解决了此类方程的求解问题.     

求解一类非线性整数规划的新方法

针对一类非线性整数规划问题，通过构造直接高散搜索方向提出了一种十分有效的算法。     

求解非线性方程的蛛网-迭代算法

用蛛网迭代算法求解非线性方程,只要求函数在定义域内存在反函数;由定理及其证明可知,不动点迭代是该迭代方法的特殊情况;通过数值实验进一步证明了该方法的有效性和实用性。     

求一类非线性偏微分方程精确解的简化试探函数法

利用试探函数法,将一个难于求解的非线性偏微分方程化为一个易于求解的代数方程,然后用待定系数法确定相应的常数,简洁地求得了一类非线性偏微分方程的精确解．将此方法应用到Burgers方程、KdV方程和KdV—Burgers方程,所得结果与已有结果完全吻合．本方法可望进一步推广用于求解其它非线性偏微分方程．     

求解非线性方程重根的区间牛顿法

讨论了求解非线性方程重根问题,针对此时Moore区间牛顿法不再适用,以及Hansen改进的区间牛顿法收敛速度慢的情况,通过引入原方程的一种相关方程,建立了求解非线性方程重根的区间牛顿法;证明了其局部平方收敛的性质,给出了数值算例。验证了新算法比Hansen改进的区间牛顿法具有更快的收敛速度,且算法是有效和可靠的。     

A matrix-free quasi-Newton method for solving large-scale nonlinear systems

One of the widely used methods for solving a nonlinear system of equations is the quasi-Newton method. The basic idea underlining this type of method is to approximate the solution of Newton’s equation by means of approximating the Jacobian matrix via quasi-Newton update. Application of quasi-Newton methods for large scale problems requires, in principle, vast computational resource to form and store an approximation to the Jacobian matrix of the underlying problem. Hence, this paper proposes an approximation for Newton-step based on the update of approximation requiring a computational effort similar to that of matrix-free settings. It is made possible by approximating the Jacobian into a diagonal matrix using the least-change secant updating strategy, commonly employed in the development of quasi-Newton methods. Under suitable assumptions, local convergence of the proposed method is proved for nonsingular systems. Numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with Newton’s, Chord Newton’s and Broyden’s methods.     

A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients

We propose a compact split-step finite difference method to solve the nonlinear Schrödinger equations with constant and variable coefficients. This method improves the accuracy of split-step finite difference method by introducing a compact scheme for discretization of space variable while this improvement does not reduce the stability range and does not increase the computational cost. This method also preserves some conservation laws. Numerical tests are presented to confirm the theoretical results for the new numerical method by using the cubic nonlinear Schrödinger equation with constant and variable coefficients and Gross-Pitaevskii equation.     

A new trust-region method with line search for solving symmetric nonlinear equations

A new trust-region method is proposed for symmetric nonlinear equations. In this given algorithm, if the trial step is unsuccessful, one line search will be used instead of repeatedly solving the subproblem of the normal trust-region method. Moreover, the global convergence is established under mild conditions by a new way. The quadratic convergence of the presented method is also proved. Numerical results show that the method is interesting for the given problems.     

A new cubic convergent method for solving a system of nonlinear equations

We present a new cubic convergent method for solving a system of nonlinear equations. The new method can be viewed as a modified Chebyshev's method in which the difference of Jacobian matrixes replaces three order tensor. Therefore, the new method reduces the storage and computational cost. The new method possesses the local cubic convergence as well as Chebyshev's method. A rule is deduced to ensure the descent property of the search direction, and a nonmonotone line search technique is used to guarantee the global convergence. Numerical results indicate that the new method is competitive and efficient for some classical test problems.     

Legendre wavelets method for solving fractional integro-differential equations

In this paper, based on the constructed Legendre wavelets operational matrix of integration of fractional order, a numerical method for solving linear and nonlinear fractional integro-differential equations is proposed. By using the operational matrix, the linear and nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which are solved through known numerical algorithms. The upper bound of the error of the Legendre wavelets expansion is investigated in Theorem 5.1. Finally, four numerical examples are shown to illustrate the efficiency and accuracy of the approach.     

竞选优化算法求解非线性方程组的应用研究

针对非线性方程组的求解在工程上具有广泛的实际意义,经典的数值求解方法存在其收敛性依赖于初值而实际计算中初值难确定的问题,将复杂非线性方程组的求解问题转化为函数优化问题,引入竞选优化算法进行求解。同时竞选优化算法求解时无需关心方程组的具体形式,可方便求解几何约束问题。通过对典型非线性测试方程组和几何约束问题实例的求解,结果表明了竞选优化算法具有较高的精确性和收敛性,是应用于非线性方程组求解的一种可行和有效的算法。